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一、基本概念
1、大圆：半径为球体半径的圆
2、球面直线：过球面上两点A、B的大圆叫做过A、B两点的球面直线
3、球面距离：以A、B为端点的大圆劣弧的长度
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1、大圆：半径为球体半径的圆
2、球面直线：过球面上两点A、B的大圆叫做过A、B两点的球面直线
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一、基本概念
1、大圆：半径为球体半径的圆
2、球面直线：过球面上两点A、B的大圆叫做过A、B两点的球面直线
3、球面距离：以A、B为端点的大圆劣弧的长度
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                                <h2>
                                    3.2 球面三角基础知识
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                                <span class="article-info">
                                    2023-10-28, 633 words, 3 min read
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                                        <p>如果没有特别说明，所讨论的球面都是单位球面</p>
<h2 id="一-基本概念">一、基本概念</h2>
<p>1、大圆：半径为球体半径的圆</p>
<p>2、球面直线：过球面上两点A、B的大圆叫做过A、B两点的球面直线</p>
<p>3、球面距离：以A、B为端点的大圆劣弧的长度</p>
<p>4、对径点：与球心同一直线的两个点（特殊的对径点：极点、极线）</p>
<p>5、球面角：从球面上A点出发的两个大圆半弧所构成的图形</p>
<p>6、球面角大小：根据边的二面角的大小度量，等于顶点的极线夹在两边之间的弧长PQ，即球心角<em>∠POQ</em>的值或两圆弧在顶点处的两切线所夹的角度∠<em>DAE</em>。</p>
<img src="http://cos.pansis.site/202310280133462.png/abc123" alt="image-20231028013338357" style="zoom:50%;" />
<p>7、球面三角形：球面上不在同一大圆上的三个点<em>A</em>、<em>B</em>、<em>C</em>，任两个都不是对径点，由连接<em>A</em>、<em>B</em>、<em>C</em>的三条大圆劣弧组成的图形。</p>
<img src="http://cos.pansis.site/202310280137481.png/abc123" alt="image-20231028013757444" style="zoom:50%;" />
<p>8、导航三角形：由地球表面上的北极点pa、观测者位置A和天体投影点b作为顶点，相应大圆劣弧作为其边，构成导航三角形<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>A</mi><msub><mi>p</mi><mi>n</mi></msub><mi>b</mi></mrow><annotation encoding="application/x-tex">ΔAp_nb</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">Δ</span><span class="mord mathdefault">A</span><span class="mord"><span class="mord mathdefault">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">b</span></span></span></span></p>
<p>9、天文三角形：由天球表面上天北极<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>P</mi><mi>N</mi></msub></mrow><annotation encoding="application/x-tex">P_N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，天体的求新投影点<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.05017em;">B</span></span></span></span>和天顶<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>Z</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">Z_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>作为顶点，和相应大圆劣弧构成的图形，称为天文三角形$Δ Z_AP_NB $</p>
<img src="http://cos.pansis.site/202311051921041.png/abc123" alt="image-20231105192121815" style="zoom: 33%;" />
<h2 id="二-球面三角形的性质">二、球面三角形的性质</h2>
<p>如果没有特别说明，所讨论的球面都是单位球面</p>
<p>边为a,b,c（用角度来表示），角为A,B,C</p>
<h4 id="1-边的基本性质">1、边的基本性质</h4>
<ul>
<li>球面三角形的三边之和 $ 0&lt; s&lt;2\pi $ （<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>2</mn><mi>π</mi></mrow><annotation encoding="application/x-tex">2\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span></span></span></span>是三点位于同一球面直线上时）</li>
<li>两边之和大于第三边,两边之差小于第三边。</li>
</ul>
<h4 id="2-角的基本性质">2、角的基本性质</h4>
<ul>
<li>
<p>三角之和 $ \pi&lt; s&lt;3\pi $  （<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>3</mn><mi>π</mi></mrow><annotation encoding="application/x-tex">3\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">3</span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span></span></span></span>是三点位于同一球面直线上时）</p>
</li>
<li>
<p>球面三角形两角之和减去第三角小于<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">π</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">π</span></span></span></span></p>
</li>
</ul>
<h4 id="3-球面三角形的公式">3、球面三角形的公式</h4>
<p>1、正弦公式：<img src="http://cos.pansis.site/202310280143298.png/abc123" alt="image-20231028014303246" style="zoom:33%;" /></p>
<p>2、边的余弦式：<img src="http://cos.pansis.site/202310280145968.png/abc123" alt="image-20231028014525927" style="zoom:33%;" /></p>
<p>3、角的余弦公式：<img src="http://cos.pansis.site/202310280145456.png/abc123" alt="image-20231028014535413" style="zoom:33%;" /></p>
<p>4、边的五元素公式：<img src="http://cos.pansis.site/202310280146165.png/abc123" alt="image-20231028014639124" style="zoom:33%;" /></p>
<p>5、角的五元素公式：<img src="http://cos.pansis.site/202310280146115.png/abc123" alt="image-20231028014642083" style="zoom:33%;" /></p>
<p>6、球面直角三角形，设C为直角，</p>
<img src="http://cos.pansis.site/202310280149545.png/abc123" alt="image-20231028014948504" style="zoom:50%;" />
<img src="http://cos.pansis.site/202310280158436.png/abc123" alt="image-20231028015656144" style="zoom:33%;" />
<img src="http://cos.pansis.site/202310280158415.png/abc123" alt="image-20231028015702252" style="zoom:33%;" />
<p>1、导航三角形、天文三角形定义</p>
<p>2、球面三角形的三边三角之和范围</p>
<p>3、球面三角形的公式</p>
<p><strong>正弦公式、边的余弦公式、角的余弦公式、边的五元素公式、角的五元素公式、球面直角三角形，设C为直角</strong></p>
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